I am having a problem with understanding the proof (proposition 2.1, page 2): https://users.renyi.hu/%7Egyarfas/Cikkek/34_GyarfasLehelTuza_ClumsyPackingOfDominoes.pdf I don't think you wanna use the link so I'll retell some first steps: You have an n by n chessboard and you place dominoes in it. Each domino covers two cells on the board with neighboring sides. You can't place dominoes on top of the others. The board is full when you can't place any more dominoes. So the question is to find the minimum number of dominoes to make the n by n board full.
The proof is for proposition 2.1, which tells us that the amount of holes (squares uncovered by dominoes on the full board, or H) does not exceed the amount of dominoes on the full board or D. (H≤D).
Then denote by D (top) and D (bot) the number of dominoes which meet the top row and the bottom row of the board, respectively. Let W(top) and W(bot) be the number of dominoes whose top side and whose bottom side touches only domino(es), respectively. So:
D = W(top) + D(top) + D' (1)
D = W(bot) + D(bot) + D'' (2)
Where D' and D'' are the amounts of dominoes not yet counted. Denote by H(top) and H(bot) the number of holes in the top row and in the bottom row of the board, respectively:
H = H(bot) + H' (3)
H = H(top) + H'' (4)
Observe that any hole not in the bottom row of the board touches the top side of some domino; furthermore, there is no domino whose top side touches two holes. Consequently, there is a bijection from the set of holes not in the bottom row into the set of all dominoes counted in D', so:
H' = D' (5)
H'' = D'' (6)
Same idea about (6). Using (1) - (6) we get (7):
D - H = W(top) + D(top) + D' - H(bot) - H'
D - H = W(bot) + D(bot) + D'' - H(top) - H''
2(D - H) = W(top) + D(top) + D' - H(bot) - H' + W(bot) + D(bot) + D'' - H(top) - H''
Remember (5) and (6), so:
2(D - H) = W(top) + D(top) - H(bot) + W(bot) + D(bot) - H(top) (8)
And also (explanation below):
D(top) - H(top) ≥ -1 (9)
D(top) - H(top) ≥ -1 (10)
Because when you have a 1 by N field (that's basically top and bottom rows) in the situation with this pattern: Hole, Domino, Hole, Domino... Hole; you can't add hole because it will touch any other hole and you will be able to place a domino here. You can have less holes and more dominoes though.
So using (8), (9) and (10) you get:
D - H ≥ (W(top) + W(bot))/2 - 1 (11)
(Where you have equality in (11) if there's equality in (9) and (10)).
And then i see the following statement, arising from (11): furthermore
W(bot) = W(top) = 0 (12)
I have no idea why is it so. If you understand, please help me. Sorry for bad english or maybe I have done something wrong; I am new there. Thank you ♥
